Virtual Equilibrium Research Articles

Non-smooth dynamics of a fishery model with a two-threshold harvesting policy

The non-linear dynamical systems theory helps implement regulatory measures to control the growth and evolution of various populations. While invasion by alien fish species is an emerging threat to native fish species in marine ecosystems, a suitable fishery management protocol needs to be incorporated in marine protected areas (MPAs) to mitigate the problem. We propose a policy of selective harvesting whenever the density of a species crosses a predefined threshold. An ecosystem with such a harvesting policy is modelled by a piece-wise smooth prey-predator type fishery model, where a control function defined by two thresholds evokes signals to cease or commence harvesting. This work reports the dynamics and bifurcations of a Filippov system, including the possibility of sliding mode dynamics. We show that the variations in the growth rate and the harvesting rate of the invasive fish species can lead to hysteresis through saddle–node and transcritical bifurcations and the appearance of a stable homoclinic orbit. We obtain the conditions for sliding domains, regular or virtual equilibria, boundary equilibria, pseudo-equilibria, and tangent points. Depending on the parameter choice, the system can stabilize at a regular equilibrium, a pseudo-equilibrium, or a pseudo-attractor, exhibiting multistability. We report the occurrence of regular or virtual equilibrium bifurcation, boundary node bifurcation and pseudo-saddle–node bifurcation. Our findings indicate that proper harvesting strategies can prevent the proliferation of invasive fish species in MPAs.

An aerodynamic optimization method based on virtual Nash equilibrium for global and local parameter decoupling

The parameterization of aerodynamic shape in the aircraft design optimization usually includes both global and local configuration features. The global parameters determine the wing span, aspect ratio, angle of attack, twist angle, and sweep angle. Meanwhile, the local parameters define the geometric shape of the wing section, including thickness and camber, etc. The changes in different types of parameters during the optimization process will lead to perturbations in the corresponding wavelength of the geometric shape, which in turn will cause different spatial frequency responses in the flow field. Generally, local parameter changes will generate high-frequency perturbations in the flow field and propagate only near the disturbance source; While global parameter changes will cause low-frequency perturbations in the flow field and propagate throughout the entire flow field. If these two different types of parameters are optimized at all once, it will lead to significant mutual interference, thereby reducing optimization efficiency. Therefore, this paper proposes an efficient optimization algorithm NashGL based on the concept of virtual Nash equilibrium. It decouples global and local variables and optimizes them simultaneously on different populations, effectively isolating interference between these two types of design variables. On the other hand, due to the small number of global variables, their rapid convergence can guide local variables towards the optimal solution. Through two aerodynamic optimization cases of NACA0012 airfoil and wing body aircraft, it is shown that compared with the method of simultaneously optimizing global and local parameters, the NashGL algorithm in this paper has the highest optimization efficiency, with an improvement of about 30% and 60%, respectively.

Limit Cycles of a Planar Piecewise Linear System with an Improper Node

This paper is concerned with the number of limit cycles for a planar piecewise linear (PWL) system with two zones separated by a straight line. Assume that one of the subsystems of the PWL system has an improper node. The number of limit cycles for saddle-improper node type, focus-improper node type and center-improper node type (the focus or the center is a virtual or boundary equilibrium) are studied. First, we introduce displacement functions and study the number of zeros of displacement functions for different types. Then, we give the parameter regions where the exact number of limit cycles is one or two (at least two) for different types.

A Filippov type predator prey system with selective harvesting of predator

In this paper, piece-wise smooth predator prey Filippov system to describe the role of refuge to prey and additional food to predators is explored. Logistic growth of prey and Holling type II predation is considered. Selective proportional harvesting of predators is included. Refuge is provided to a fraction of preys when their density is below a threshold value. This reduces the food availability for predators. To compensate this, additional food is provided to predators. The dynamics on the boundary is computed using the Filippov regularization method. The stability and existence of different equilibrium states of the independent smooth subsystems are analyzed. The role of quality of additional food with respect to prey in the characterization of the equilibrium points as regular or virtual equilibrium points for the Filippov system is investigated. It is verified that if only refuge is provided to prey with no additional food to predators then both the interior equilibrium state will never be regular simultaneously for the Filippov system. It is observed from the sliding mode dynamics that there is no escaping region. It is proved that both refuge to prey and additional food to predators enhances the sliding region on the boundary. The analysis of boundary equilibrium points, tangent points and pseudoequilibrium points on the boundary is carried out. The role of effort on harvesting on the expansion or shrinking of the limit cycle is elaborated numerically. Boundary bifurcation for the interior equilibrium state of the subsystems is demonstrated. Numerical simulation is used to elaborate the dynamics of Filippov system.

Stable visually guided reaching does not require an internal feedforward model to compensate for internal delay: Data and model

Visually guided reaches are performed in ≈1s. Given unstable feedback control with neural transmission delay, stable visually guided reaching is assumed to require internal feedforward models that generate simulated feedback without delay that combines with actual feedback for stability. We investigated whether stable visually guided reaching requires internal models to handle such delay. Participants performed rapid targeted reaches in a virtual environment with different mappings between speeds of the hand and hand avatar. First, participants reached with visual guidance and constant mapping. Second, feedforward reaches were performed with constant mapping and hand avatar only visible at reach start and end. Reaches were accurate. Third, participants performed reaches with visual guidance and different mappings every trial. We expected performance as in the first condition. Finally, feedforward reaches with variable mapping yielded large errors showing visual guidance in the previous condition was successful despite an ineffective internal model. We simulated reaches using a proportional rate model with disparity Tau controlling the virtual Equilibrium Point in an Equilibrium Point (EP) model. The time dimensioned information and dynamic remained stable with delayed feedback. Finally, we fit movement times using the proportional rate EP model with 0msec, 50msec, and 100msec delay. With the fitted model parameters, we compared the model reach trajectories with the behavioral trajectories. Stable visually guided reaching did not require an internal feedforward model.

‘Binary’ Entrepreneurs, ‘Digital’ Macroeconomics and the ‘Virtual World’

This paper examines fundamental concepts of entrepreneurship and macroeconomics in the age the virtual world as digital socio-economics takes center stage. A model is developed and advanced. It argues that the virtual world system equilibrium will require certain physical and virtual conditions to obtain the ideal socio-economic system parameters. Furthermore, it alludes that e-resources, human capital, e-material and e-financial resources gained by prudence, is the exact extent to which the entire ecosystem can will expand without crisis, disruptions and breakdowns. This virtual world in which humans will soon populate will be predesigned with built-in macroeconomic shock-absorbing sub-systems to ensure prosperity. Keywords: ICT Economics, Entrepreneurship, Virtual World JEL Classification: 03 DOI: 10.7176/IKM/12-5-02 Publication date: September 30 th 2022

Modeling the influence of health education on individual behavior in the process of disease transmission

Health education directly affects the different behaviors of individuals who visit and do not visit after illness by reporting the number of patients and their changing rate. As a result, visiting and non‐visiting patients have different infection rates, cure rates, and disease‐related mortality. In response to this impact, this article establishes a model to further analyze the impact of health education on the spread of diseases. We divide patients into visiting and non‐visiting patients and introduce a weighting function that depends on the number of visiting patients and their changing rate. We constructs an implicit differential equations and use the properties of Lambert W function and its inverse function to transform it into piecewise smooth systems with explicit definitions. According to the analysis method of switched system, the two subsystems are analyzed qualitatively. The results show that health education cannot affect the basic reproduction number of disease transmission, but it causes the system to produce many types of equilibria. In addition, the difference in the healing rate and self‐healing rate between visiting and non‐visiting patients will affect the number of visiting and non‐visiting patients and the stability of the equilibria. Moreover, the system will generate fold branch. In addition, the patient's healing/self‐healing rate and the implementation of health education will affect the appearance of virtual or false equilibrium. Numerical simulation and sensitivity analysis further confirm that the total number of visiting and non‐visiting patients will decrease with the increase of health education. Finally, our research results show that health education can reduce the infection rate only if there are enough susceptible people receiving health education.

Complex dynamics in a Filippov pest control model with group defense

In this paper, an integrated pest management Filippov model with group defense behavior is established, which takes the population density of pests as the control index of integrated pest management. First, under the condition that both subsystems have a globally asymptotically stable equilibrium, the dynamics of the established model are systematically analyzed, including the sliding mode dynamics, the existence and global stability of the real, virtual and pseudo equilibrium, as well as boundary-node and boundary-focus bifurcation. Next, we study the complex dynamics in the Filippov model when an unstable node (focus) or a stable limit cycle occurs in the subsystem by using numerical simulations. The results show that although there are no closed orbits in subsystem, a stable periodic solution may exist for Filippov system after switching perturbation. Finally, we conclude that the group defense behavior of pest makes it harder to control.

Brain Processes Involved in Motor Planning Are a Dominant Factor for Inducing Event-Related Desynchronization.

Event-related desynchronization (ERD) is a relative attenuation in the spectral power of an electroencephalogram (EEG) observed over the sensorimotor area during motor execution and motor imagery. It is a well-known EEG feature and is commonly employed in brain-computer interfaces. However, its underlying neural mechanisms are not fully understood, as ERD is a single variable correlated with external events involving numerous pathways, such as motor intention, planning, and execution. In this study, we aimed to identify a dominant factor for inducing ERD. Participants were instructed to grasp their right hand with three different (10, 25, or 40%MVF: maximum voluntary force) levels under two distinct experimental conditions: a closed-loop condition involving real-time visual force feedback (VF) or an open-loop condition in a feedforward (FF) manner. In each condition, participants were instructed to repeat the grasping task a certain number of times with a timeline of Rest (10.0 s), Preparation (1.0 s), and Motor Execution (4.0 s) periods, respectively. EEG signals were recorded simultaneously with the motor task to evaluate the time-course of the event-related spectrum perturbation for each condition and dissect the modulation of EEG power. We performed statistical analysis of mu and beta-ERD under the instructed grasping force levels and the feedback conditions. In the FF condition (i.e., no force feedback), mu and beta-ERD were significantly attenuated in the contralateral motor cortex during the middle of the motor execution period, while ERD in the VF condition was maintained even during keep grasping. Only mu-ERD at the somatosensory cortex tended to be slightly stronger in high load conditions. The results suggest that the extent of ERD reflects neural activity involved in the motor planning process for changing virtual equilibrium point rather than the motor control process for recruiting motor neurons to regulate grasping force.

Non-smooth dynamics emerging from predator-driven discontinuous prey dispersal

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey–predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov’s regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics.

Rich dynamics of a Filippov avian-only influenza model with a nonsmooth separation line

Depopulation of birds has been authenticated to be an effective measure in controlling avian influenza transmission. In this work, we establish a Filippov avian-only model incorporating a threshold policy control. We choose the index—the maximum between the infected threshold level I_{T} and the product of the number of susceptible birds S and a ratio threshold value ξ—to decide on whether to trigger the control measures or not, which then leads to a discontinuous separation line and two pieces of sliding-mode domains. Meanwhile, one more sliding-mode domain gives birth to more complex dynamics. We investigate the global dynamical behavior of the Filippov model, including the real and/or virtual equilibria and the two sliding modes and their dynamics. The solutions will eventually stabilize at the real endemic equilibrium of the subsystem or the pseudoequilibria on the two sliding modes due to different threshold values. Therefore an effective and efficient threshold policy is essential to control the influenza by driving the number of infected birds below a certain level or at a previously given level.

Synchronous Glacial Cycles in a Nonsmooth Conceptual Climate Model with Asymmetric Hemispheres

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 6 January 2021Accepted: 07 September 2021Published online: 14 December 2021Keywordsnonsmooth dynamical systems, virtual equilibria, nonsmooth return map, ice--albedo feedback, paleoclimate, glacial cyclesAMS Subject Headings49J52, 37N99, 86A40Publication DataISSN (online): 1536-0040Publisher: Society for Industrial and Applied MathematicsCODEN: sjaday

Analysis of a piecewise linear aeroelastic system with and without tuned vibration absorber

The dynamics of a two-degrees-of-freedom (pitch–plunge) aeroelastic system is investigated. The aerodynamic force is modeled as a piecewise linear function of the effective angle of attack. Conditions for admissible (existing) and virtual equilibria are determined. The stability and bifurcations of equilibria are analyzed. We find saddle-node, border collision and rapid bifurcations. The analysis shows that the pitch–plunge model with a simple piecewise linear approximation of the aerodynamic force can reproduce the transition from divergence to the complex aeroelastic phenomenon of stall flutter. A linear tuned vibration absorber is applied to increase stall flutter wind speed and eliminate limit cycle oscillations. The effect of the absorber parameters on the stability of equilibria is investigated using the Liénard–Chipart criterion. We find that with the vibration absorber the onset of the rapid bifurcation can be shifted to higher wind speed or the oscillations can be eliminated altogether.

Fractional-Order Hidden Attractor Based on the Extended Liu System

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

A Holling Type II Discrete Switching Host-Parasitoid System with a Nonlinear Threshold Policy for Integrated Pest Management

A nonlinear discrete switching host-parasitoid model with Holling type II functional response function, in which the switch is guided by an economic threshold (ET), is proposed. Thus, if the weighted density of two generations of the host population increases and exceeds the ET, then integrated pest management (IPM) measures are enacted, i.e., biological and chemical measures are implemented together, assuming that the chemical immediately precedes the biological inputs to avoid pesticide-induced deaths of the natural enemies. First, the existence and local stability of the equilibria of two subsystems were studied, and the existence and coexistence of several types of equilibria of a nonlinear switching system were analysed. Next, the nonlinear switching system was investigated by numerical simulation, showing that the system exhibits quite complex dynamic behaviour. A two-dimensional bifurcation diagram revealed the existence and coexistence regions of different types of equilibria including regular and virtual equilibria. Moreover, period-adding bifurcations in two-dimensional parameter spaces were found. One-dimensional bifurcation diagrams revealed that the system has periodic, quasiperiodic, and chaotic solutions, Neimark–Sacker bifurcation, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the initial densities of hosts and parasitoids associated with host outbreaks and their biological implications are discussed.

A 3D direct coupling method for steady ship hydroelastic analysis

Asides from the influence of incoming waves, ships can experience steady motions, such as rigid-body sinkage and trim motions, and flexible-body vertical bending motions, due to a constant forward speed even under calm water conditions. In this paper, a novel approach to analyze steady-ship hydroelasticity, particularly for the steady-ship motions and surrounding steady-wave disturbances, is proposed using a three-dimensional (3D) direct coupling method, based on a higher-order boundary element method (HOBEM) and a higher-order shell finite element method (FEM). Within the linearized framework, a solution method is proposed based on a two-step procedure, using two types of Neumann–Kelvin (NK) linear flow models for the fluid part and a virtual work equilibrium equation for the structural part. The first step is to compute a mean position wave-resistance problem using the modified NK equation, the second step is to solve a perturbed position wave-resistance problem, by employing a classical NK model and a virtual work equation based on the first step’s solution. Detailed mathematical formulation and numerical procedures are described, and a few numerical results are illustrated. These include both rigid and flexible steady-ship motions, Von-Mises stress distributions, and wave-resistance coefficients for Froude numbers ranging from 0.15 to 0.5. Furthermore, the numerical results obtained using the present direct coupling method and a modal-based one are compared.

Speed Ripple Reduction of Direct-Drive PMSM Servo System at Low-Speed Operation Using Virtual Cogging Torque Control Method

In this article, a virtual cogging torque (VCT) control method to reduce the speed ripple of direct-drive permanent magnet synchronous machine servo system under low-speed conditions is proposed. Compared with other factors, at low speeds, the cogging torque is the main factor that deteriorates the drive performance, even induces speed oscillations. Especially in this article, due to volume limitation, the cogging torque is designed larger than normal one in order to remove the need of brake. Based on the model of permanent magnet synchronous motor, the cause and effect of the cogging torque are analyzed. Inspired by the characteristic of cogging torque, the VCT control method is proposed and investigated to significantly reduce the speed ripple at low speeds. The main idea of this proposed control method is to produce a proper VCT and continuously move the corresponding virtual stable equilibrium point to drive the rotor smoothly. In addition to the principle of this control method, its analysis and implementation are studied as well. Simulation and experimental results from the prototype demonstrate that the proposed control method is correct and valid, and it is simple and effective to smooth the speed at low-speed operations.

Considerations for improved performance of competition association assays analysed with the Motulsky-Mahan's "kinetics of competitive binding" model.

Background and PurposeTarget engagement dynamics can influence drugs' pharmacological effects. Kinetic parameters for drug:target interactions are often quantified by evaluating competition association experiments—measuring simultaneous protein binding of labelled tracers and unlabelled test compounds over time—with Motulsky–Mahan's “kinetics of competitive binding” model. Despite recent technical improvements, the current assay formats impose practical limitations to this approach. This study aims at the characterisation, understanding and prevention of these experimental constraints, and associated analytical challenges.Experimental ApproachMonte Carlo simulations were used to run virtual kinetic and equilibrium tracer binding and competition experiments in both normal and perturbed assay conditions. Data were fitted to standard equations derived from the mass action law (including Motulsky–Mahan's) and to extended versions aiming to cope with frequently observed deviations of the canonical traces. Results were compared to assess the precision and accuracy of these models and identify experimental factors influencing their performance.Key ResultsKey factors influencing the precision and accuracy of the Motulsky–Mahan model are the interplay between compound dissociation rates, measurement time and interval frequency, tracer concentration and binding kinetics and the relative abundance of equilibrium complexes in vehicle controls. Experimental results produced recommendations for better design of tracer characterisation experiments and new strategies to deal with systematic signal decay.Conclusions and ImplicationsOur data advances our comprehension of the Motulsky–Mahan kinetics of competitive binding models and provides experimental design recommendations, data analysis tools, and general guidelines for its practical application to in vitro pharmacology and drug screening.

Threshold Strategy for Nonsmooth Filippov Stage-Structured Pest Growth Models

In order to control pests and eventually maintain the number of pests below the economic threshold, in this paper, based on the nonsmooth dynamical system, a two-stage-structured pest control Filippov model is proposed. We take the total number of juvenile and adult pest population as the control index to determine whether or not to implement chemical control strategies. The sliding-mode domain and conditions for the existence of regular and virtual equilibria, pseudoequilibrium, boundary equilibria, and tangent points are given. Further, the sufficient condition of the locally asymptotic stability of pseudoequilibrium is obtained. By numerical simulations, the local bifurcations of the equilibria are discussed. Our results show that the total number of pest populations can be successfully controlled below the economic threshold by taking suitable threshold policy.

Visible light spectroscopic analysis of Methylene Blue in water; the resonance virtual equilibrium hypothesis

After a critical approach to the centennial literature on Methylene Blue self-aggregation in water, accurate visible-light absorption spectra and thermodynamic equilibrium constants for the monomeric and dimeric forms of this dye in water at different temperatures were obtained by means of a novel procedure based on the Beer-Lambert equation. A thermo-solvatochromic effect was clearly detected in the monomer spectra, which we believe to be the main source of inaccuracy in all previous works. The only published explanation for this phenomenon, the existence of a colorless form of the monomer, is refuted here. Instead, a new notion has been introduced; the resonance virtual equilibrium hypothesis. According to this hypothesis the temperature-dependent absorption behavior of the monomer is provoked by the change in its electron charge distribution with the variation in the temperature-dependent dielectric constant of water. The monomer charge distribution stands between those of the virtual resonance forms, being the absorption spectrum of the monomer a composition of the theoretical spectra for the virtual mesomers, whose proportion is established by a temperature-dependent equilibrium constant. The application of this hypothesis allowed the virtual equilibrium constants and spectra for the mesomers to be evaluated with remarkable accuracy.